Restriction of intramuscular neurite branching and extension is lost in agrin and rapsyn-deficient mice

The phrenic nerve enters the diaphragm at approximately embryonic day 12.5 (E12.5) in the mouse. The secondary nerve trunk advances along the centre of the diaphragm muscle and extends tertiary branches primarily towards the lateral side during normal embryonic development. In the present study we quantified the intramuscular neurite branching in the most ventral region of the diaphragm at E15.5 and E18.5 in wild-type mice, agrin knock-out mice (KOAG) and rapsyn knock-out mice (KORAP). KOAG and KORAP have decreased muscle contraction due to their inability to maintain/form acetylcholine receptor (AChR) clusters during embryonic development. Heterozygote mothers were anaesthetised via an overdose of Nembutal (30 mg; Boeringer Ingelheim, Ridgefield, CT, USA) and killed via cervical dislocation. There were increases in the number of branches exiting the medial side of the phrenic nerve trunk in KOAG and KORAP compared to wild-type mice, but not on the lateral side at E15.5 and E18.5. However, the number of bifurcations in the periphery significantly increased on both the medial and lateral sides of the diaphragm at E15.5 and E18.5 in KOAG and KORAP compared to control mice. Furthermore, neurites extended further on both the medial and lateral sides of the diaphragm at E15.5 and E18.5 in KOAG and KORAP compared to wild-type mice. Together these results show that the restriction of neurite extension and bifurcations from the secondary nerve trunk is lost in both KOAG and KORAP allowing us the opportunity to investigate the factors that restrict motoneuron behaviour in mammalian muscles.

On the Maximum of a Branching Process Conditioned on the Total Progeny

The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.